1. business and industrial
2. chemicals industry
3. dyes and pigments

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,
1
2
Surfzone to inner-shelf exchange estimated from dye
tracer balances
K. Hally-Rosendahl, Integrative Oceanography Division, Scripps Institution of Oceanography,
UCSD, 9500 Gilman Dr., La Jolla CA 92093-0209 ([email protected])
F. Feddersen, Integrative Oceanography Division, Scripps Institution of Oceanography, UCSD,
9500 Gilman Dr., La Jolla CA 92093-0209 ([email protected])
D. B. Clark, Woods Hole Oceanographic Institution, MS#12, 266 Woods Hole Rd., Woods
Hole, MA 02543 ([email protected])
R. T. Guza, Integrative Oceanography Division, Scripps Institution of Oceanography, UCSD,
9500 Gilman Dr., La Jolla CA 92093-0209 ([email protected])
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Abstract.
Surfzone and inner-shelf tracer dispersion are observed at an approximately
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alongshore-uniform beach. Fluorescent Rhodamine WT dye tracer, released
6
near the shoreline continuously for 6.5 h, is advected alongshore by break-
7
ing wave- and wind-driven currents, and ejected offshore from the surfzone
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to the inner-shelf by transient rip currents. Aerial multispectral imaging of
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dye concentration and in situ measurements of dye, waves, and currents pro-
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vide tracer transport and dilution observations spanning approximately 400
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m cross-shore and 3 km alongshore. The alongshore dilution of near-shoreline
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dye follows weak power-law decay; a 10-fold increase in downstream distance
13
from the source only reduces the concentration by 50%. Surfzone and inner-
14
shelf dye mass balances close, and in 5 h roughly 1/2 of the surfzone-released
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dye is transported offshore to the inner-shelf. Observed cross-shore transports
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from the surfzone to the inner-shelf are parameterized well using a bulk ex-
17
change velocity (correlation r2 = 0.85 and best fit slope 0.7). The best fit
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cross-shore exchange velocity u∗ ≈ 1 × 10−2 m s−1 is similar to u∗ es-
19
timated using temperature observations on another day with similar wave
20
conditions, confirming the dominance of transient rip currents in surfzone
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to inner-shelf cross-shore exchange during moderate waves at an alongshore-
22
uniform beach.
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1. Introduction
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The nearshore region, consisting of the surfzone (shoreline to xb , the seaward bound-
24
ary of depth-limited wave breaking) and the inner-shelf (xb to approximately 20 m water
25
depth), is vitally important to coastal economies, recreation, and human and ecosystem
26
health. However, nearshore water quality is often compromised by terrestrial runoff and
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offshore waste disposal [e.g., Koh and Brooks, 1975; Schiff et al., 2000; Halpern et al., 2008].
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Globally, microbial pathogen exposure from polluted nearshore water causes an estimated
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120 million gastrointestinal illnesses and 50 million severe respiratory illnesses annually
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[Dorfman and Stoner , 2012] with significant economic impacts. Furthermore, excess nutri-
31
ents in polluted runoff can spur rapid growth of harmful algal blooms (HABs), damaging
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ecosystems and causing serious and even life-threatening human illnesses through direct
33
ocean exposure or consumption of algal-contaminated seafood [Dorfman and Haren, 2013].
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Pathogens, nutrients, HABs, and other contaminants all act as nearshore tracers, the
35
transport and dilution of which are governed by surfzone and inner-shelf physical pro-
36
cesses. Yet, despite the detriment of contaminated coastal water to our health and econ-
37
omy, understanding of nearshore tracer transport and mixing remains relatively poor.
38
Several field experiments [e.g., Harris et al., 1963; Inman et al., 1971; Grant et al., 2005]
39
have tracked near-shoreline-released dye, simulating point source tracers of terrestrial or
40
surfzone origin. However, these studies were limited by sparse sampling and/or small spa-
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tiotemporal extents of quantitative observations. A rapid-sampling, jetski-based dye mea-
42
surement platform [Clark et al., 2009] provided greatly improved observations. Analyses of
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dye plume evolution at Huntington Beach, California (HB06) showed that surfzone cross-
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shore tracer dispersion is dominated by horizontal eddies [Clark et al., 2010; Feddersen
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et al., 2011; Clark et al., 2011] forced by finite crest length wave breaking [Peregrine, 1998;
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Spydell and Feddersen, 2009; Clark et al., 2012; Feddersen, 2014]. However, HB06 obser-
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vations were limited to ≤ 2 hours and usually < 400 m downstream of the dye source with
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analyses specifically restricted to surfzone-contained portions of the dye plumes. While
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shoreline-source tracers are first transported and mixed within the surfzone, their fate
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is ultimately determined by exchange with the inner-shelf [Hally-Rosendahl et al., 2014].
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An improved understanding of long distance surfzone tracer dilution requires quantitative
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estimates of net cross-shore exchange between the the surfzone and inner-shelf.
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The surfzone and inner-shelf are governed by drastically different dynamics. The sur-
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fzone is dominated by breaking-wave-driven currents [e.g., Thornton and Guza, 1986]
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and horizontal eddies [e.g., Peregrine, 1998; Clark et al., 2012], whereas the inner-shelf is
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forced by a combination of wind, tides, buoyancy, and both surface and internal waves
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[e.g., Lucas et al., 2011; Lentz and Fewings, 2012; Kumar et al., 2014; Sinnett and Fed-
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dersen, 2014]. Furthermore, the surfzone is vertically well mixed while the inner-shelf
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can be strongly stratified immediately offshore of the surfzone [Hally-Rosendahl et al.,
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2014]. The intersection of, and exchange between, these dynamically different regions is
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particularly complex. The fall 2009 IB09 experiment at Imperial Beach, California was
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designed to observe shoreline-released dye with better resolution and over longer times
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and distances than preceding studies, and particularly to quantify surfzone and inner-shelf
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exchange. Hally-Rosendahl et al. [2014] analyzed 29 Sept dye and temperature observa-
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tions across the surfzone and inner-shelf, spanning 7-12 hours and 700 m alongshore.
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Downstream dye dilution followed a power-law decay and was weaker than that expected
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for a Fickian diffusive process as observed and modeled for surfzone-contained portions
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of dye plumes [Clark et al., 2010, 2011]. Computed using temperature observations, a
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bulk cross-shore exchange velocity ≈ 1 × 10−2 m s−1 was 10-15× larger than the estimated
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Stokes drift driven exchange velocity, indicating that the observed transient rip currents
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dominated surfzone/inner-shelf cross-shore exchange at the alongshore-uniform Imperial
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Beach [Hally-Rosendahl et al., 2014]. However, the cross-shore dye transport was not mea-
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sured, as coupled dye and velocity observations at high temporal and spatial resolutions
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over large alongshore distances are required to resolve the transient and small scale rip
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current ejection structure.
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Here, aerial and in situ dye observations on 13 Oct, spanning the surfzone and inner-shelf
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for even greater alongshore distances (up to 3 km) than 29 Sept, are used to investigate
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far-downstream dye dilution and cross-shore transport estimated from surfzone and inner-
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shelf dye mass balances. The experiment site, dye release, instrument platforms, and
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sampling schemes are described in section 2. Wave and alongshore current conditions
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are presented in section 3.1. In section 3.2, aerial dye observations are described, and
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time periods and spatial regions for subsequent analyses are established. Surfzone cross-
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shore dye structure is described in section 3.3. Alongshore dye dilution and transport are
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examined in sections 3.4 and 3.5, respectively. Sections 4.1-4.3 present dye mass balances
85
(estimation methods are described in Appendix A). The closure of these balances allows
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for observational estimates of cross-shore dye transport (section 4.4) which are compared
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to parameterized estimates in section 5.1. Section 6 is a summary.
2. IB09 Experiment
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2.1. Field Site and Coordinate System
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IB09 field observations were acquired during fall 2009 at Imperial Beach, California
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(32.6◦ N, 117.1◦ W), a west (269.6◦ ) facing beach with an approximately straight shoreline
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(Figure 1). In the right-handed coordinate system, cross-shore coordinate x increases
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negatively seaward (x = 0 m at the mean shoreline), alongshore coordinate y increases
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positively toward the north (y = 0 m at the dye release location), and vertical coordinate
93
z increases positively upward (z = 0 at mean sea level). The dye release discussed here
94
took place on 13 October, 2009. Bathymetry surveys from 9 October and 19 October
95
were similar, and are averaged to give a representative bathymetry for 13 October that is
96
approximately alongshore-uniform (Figure 1). All times are in PDT.
2.2. Dye Release
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Fluorescent Rhodamine WT dye (2.1 × 108 parts per billion (ppb)) was released con-
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tinuously at 2.4 mL s−1 near the shoreline at (x, y) = (−10, 0) m for approximately 6.5 h
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(10:39-17:07 h). Visual observations suggested rapid vertical mixing, and measured dye
100
concentrations were reduced from O(108 ) ppb to O(102 ) ppb within 10 m of the release.
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Therefore, the dye specific gravity was quickly reduced from 1.2 to ≈ 1. Rhodamine WT
102
has a photochemical decay e-folding time of approximately 667 h of sunlight [e.g., Smart
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and Laidlaw , 1977]; decay over the ≈ 9 h of sunlight during this study is negligible.
2.3. In Situ Instrumentation: Surfzone and Inner-Shelf
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2.3.1. Cross-Shore Array
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A 125 m-long cross-shore array of six fixed, near-bed instrument frames (denoted f1-f6,
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onshore to offshore) was deployed from near the shoreline to approximately 4 m water
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depth (diamonds, Figure 1). The frames held Paros pressure sensors, SonTek acous-
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tic Doppler velocimeters (ADVs), Yellow Springs Instrument Company thermistors, and
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WET Labs ECO Triplet fluorometers (hereafter ET) to measure dye concentration D.
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One frame (f4), located near the seaward edge of the surfzone, held instruments at three
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different vertical locations (0.2, 0.7, and 1.3 m above the bed). Cross-shore array instru-
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ments sampled for 51 min each hour, with the remaining 9 min used by the ADVs to
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estimate bed location [Feddersen, 2012; Spydell et al., 2014]. On 13 Oct, the dye release
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(y = 0 m) was 248 m south of this cross-shore array (Figure 1).
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2.3.2. Surfzone Near-Shoreline Alongshore Array
Four thermistor-equipped ETs were deployed near the shoreline at y = 82, 546, 1069, 1662 m
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(circles, Figure 1), referred to as SA1-SA4, respectively. For some analyses, ET data from
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f2 (at yf = 248 m) are used in conjunction with data from SA1-SA4. The ET on f2
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sampled throughout (and after) the dye release, while SA1-SA4 were deployed after the
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dye release started.
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2.3.3. Cross-Shore Jetski Transects
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Surface dye concentration and temperature were measured with fluorometers and ther-
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mistors mounted on two GPS-tracked jetskis [Clark et al., 2009] that drove repeated cross-
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shore transects from x ≈ −300 m to the shoreline (e.g., Figure 1) at various designated
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alongshore locations between y = 5 m and y ≈ 2 km. The alongshore spacing between
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transects varied from approximately 20 m (near the release) to 300 m (far downstream of
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the release). Analyses only include shoreward transects, when jetskis were driven immedi-
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ately in front of bores to minimize turbidity from bubbles and suspended sand. Seaward
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transects, sometimes corrupted when jetskis swerved or became airborne jumping over
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waves, are discarded.
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2.3.4. Inner-Shelf Alongshore Boat Transects
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Offshore of the surfzone, the vertical and alongshore structure of dye concentration
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and temperature were measured with a vertical array of five thermistor-equipped ETs
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towed alongshore behind a small boat. The vertical array sampled from z = −1 to
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−3 m at 0.5 m spacing. During 14:06-17:43 h, repeated ≈ 2 km-long alongshore transects
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(e.g., Figure 1) were driven at roughly 1 m s−1 at a mean cross-shore location nominally
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twice the surfzone width. The transects were approximately shore-parallel with deviations
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to avoid large waves.
2.4. Remote Sensing Instrumentation: Inner-Shelf
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Aerial observations of near-surface dye concentration were obtained from a small plane
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with a multispectral camera system and coupled global positioning and inertial navigation
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systems [Clark et al., 2014]. Two cameras captured images near the peak excitation
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and emission wavelengths of the fluorescent Rhodamine WT. Dye concentrations were
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determined by calibrating the ratio of emission to excitation radiances with coincident
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in situ data. Aerial dye concentration errors range from ±1.5 ppb near D = 0 ppb to
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±4.5 ppb near D = 20 ppb. The georeferenced aerial images are combined into mosaics
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and regridded onto a rectangular grid with 2 m × 2 m lateral resolution. See Clark et al.
147
[2014] for details.
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Between 11:21 and 15:32 h, 23 mosaic images were obtained, each separated by roughly
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6 min (with a longer gap from 13:08 to 14:56 h). The dye field was imaged from the
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shoreline to roughly 400 m offshore and from the release to roughly 3 km downstream.
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Pixels with excitation image brightness above an empirical threshold [Clark et al., 2014]
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owing to sun glitter or white foam from breaking waves are discarded. The surfzone is
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therefore often poorly resolved, and quantitative analyses of aerial images are confined to
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the inner-shelf.
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2.4.1. Corrections to Measured Dye Fluorescence
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All aerial and in situ dye observations are corrected for temperature [Smart and Laidlaw ,
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1977], and all in situ dye observations are corrected for turbidity [Clark et al., 2009].
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Corrected D typically differ from measured D by less than 5%.
3. Observations
3.1. Wave, Wind, and Alongshore Current Conditions
159
During the dye release, the incident wave field (with peak period Tp = 13 s) is relatively
160
constant, and the tide varies less than 0.7 m (low tide at 12:33 h). The release-averaged
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significant wave height Hs (x) shoals to a maximum of 0.87 m at f4 (break point xb =
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−81 m, mean breaking depth hb = 2.1 m, Figures 2a and 2c). The mean alongshore
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current V (x) is northward (positive) at all f1-f6 locations, with a near-shoreline maximum
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of 0.40 m s−1 (Figure 2b). Offshore, V decreases to 0.12 m s−1 at the seaward surfzone
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boundary xb and then increases slightly to 0.17 m s−1 at inner-shelf f6 (x = −135 m,
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Figure 2b). The mean surfzone (f1-f4) alongshore current is VSZ = 0.22 m s−1 , and the
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mean inner-shelf (f4-f6) alongshore current is VIS = 0.14 m s−1 . Wind is from the south
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at 4-7 m s−1 .
3.2. Inner-shelf Surface Dye Evolution
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Aerial images (e.g., Figures 3a-3f) spanning 0:42-4:53 h after the t0 = 10:39 h start of the
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dye release are partitioned into three time periods based on temporal gaps in images and
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the dye plume evolution: period I (early in release, 11:21-11:53 h), period II (mid-release,
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12:08-13:01 h), and period III (late in release, 14:56-15:32 h). Approximately 40 min after
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the release begins (Figure 3a, period I), surfzone dye has advected about 600 m alongshore
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at ≈ 0.25 m s−1 , consistent with in situ VSZ = 0.22 m s−1 (Figure 2b). Surfzone dye is
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ejected onto the inner-shelf in narrow (≈ 50 m) alongshore bands (Figure 3a), presumably
176
due to transient rip currents [e.g., Hally-Rosendahl et al., 2014]. As the dye release contin-
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ues (Figures 3b-3d, period II), the leading portion of inner-shelf dye is alongshore-patchy
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with length scales ≈ 50 m (as in Figure 3a). Behind the leading edge, slower alongshore
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advection of inner-shelf dye (e.g., the feature at y ≈ 1250 m and 1500 m in Figures 3e
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and 3f, respectively, period III) is apparent at a speed of ≈ 0.15 m s−1 , consistent with
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in situ VIS = 0.14 m s−1 (Figure 2b). At these longer times and downstream distances
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(Figures 3e and 3f, period III), inner-shelf dye advects alongshore, disperses cross-shore,
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and moves to larger alongshore length scales. In particular, Figures 3e and 3f reveal a
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coherent nearshore eddy feature (at y ≈ 1250 m and 1500 m, respectively) with an along-
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shore length scale ≈ 300 m, roughly six times larger than the length scales of inner-shelf
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dye patches when recently ejected from the surfzone.
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In addition to the temporal partitioning into periods I, II, and III (defined above),
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dye is cross-shore partitioned into the surfzone and inner-shelf (separated by xb = −81 m,
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section 3.1) and alongshore partitioned into near- and far-field regions A and B, separated
190
by the cross-shore frame array at yf = 248 m (Figure 3a).
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The leading alongshore edge of the dye plume yp (t) is defined as the northernmost
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location where aerial-imaged inner-shelf D exceeds 3 ppb within 40 m of xb (green triangles
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in Figures 3a-3e). The plume leading edge yp (t) increases roughly linearly during each
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time period, with the fastest advance during period III (Figure 4). The associated mean
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alongshore velocity over periods I, II, and III is 0.17 m s−1 (Figure 4), between the surfzone
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and inner-shelf means VSZ = 0.22 m s−1 and VIS = 0.14 m s−1 .
3.3. Cross-surfzone Mean Dye Profiles
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Time-averaged surface dye profiles D(x, yj ) from repeated jetski cross-shore transects
198
at designated alongshore locations yj are cross- and alongshore-binned corresponding to
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where near-shoreline dye is released (y = 0 m) and measured (SA1, f2, SA2, SA3, and
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SA4, Figure 1). Near the release (y = 14 m), dye concentration is high (≈ 80 ppb)
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near the shoreline and decays to ≈ 10 ppb near xb (Figure 5). Immediately downstream
202
(y = 87 m), as dye is dispersed offshore, the mean dye cross-surfzone profile begins to
203
flatten. At y ≥ 207 m, dye is well mixed across the surfzone (Figure 5), indicating that
204
surfzone-representative D can be estimated with near-shoreline measurements.
3.4. Near-Shoreline Alongshore Dye Dilution
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Here, alongshore dye dilution is examined with near-shoreline in situ data from SA1-
206
SA4 and f2 (yellow circles and diamond, respectively, Figure 3). The SA1-SA4 data start
207
times (beginning progressively later with alongshore distance from the release) indicate
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SA1-SA4 deployment times instead of dye arrival times at those locations (Figure 6).
209
With the exception of SA3 (y = 1069 m), SA instruments were recovered shortly after
210
the dye release ended.
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When the dye field is roughly stationary, mean near-shoreline dye concentration decays
212
with downstream distance from the release (Figures 6 and 7) because dye is dispersed
213
cross-shore onto the inner-shelf as it is advected alongshore (Figure 3). Near-shoreline
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dye variability also decreases significantly with y (Figure 6 and vertical bars in Figure 7).
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For example, near the release at y = 248 m, D varies between 0-80 ppb with a mean of
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15 ppb, while at y = 1662 m, D varies between 4-12 ppb with a mean of 8 ppb (Figures 6b,
217
218
6e, and 7). Furthermore, the time scale of dye variability increases with downstream
−1/2
2
2
distance from the release (Figures 6a-6e); the characteristic time scale dD
/D
dt
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increases monotonically from 71 s at y = 82 m to 584 s at y = 1662 m. Together with
220
the increasing time scale, the reduction of D variability relative to the mean suggests
221
that surfzone alongshore dye length scales also increase with y, consistent with inner-shelf
222
observations (Figures 3a-3f).
The near-shoreline mean dye dilutes following a power-law,
D = D0 (y/y0 )α ,
(1)
223
where y0 = 1 m is chosen for simplicity. The best fit constants are D0 = 98 ppb and
224
α = −0.3 (dashed line in Figure 7). Similar downstream dilution structure was ob-
225
served over shorter distances (700 m) during a 29 Sept dye release with α = −0.2 [Hally-
226
Rosendahl et al., 2014]. The downstream dilution power-law exponents α = (−0.3, −0.2)
227
are smaller than that expected for a shoreline Fickian diffusive process (α = −0.5, dye flux
228
proportional to mean concentration gradient). Deviation from α ≈ −0.5 observed and
229
modeled for surfzone-contained dye plumes [Clark et al., 2010, 2011] indicates that the
230
dispersion is not Fickian after dye saturates the surfzone and spreads onto the inner-shelf
231
[Hally-Rosendahl et al., 2014]. Note that α = −0.3 corresponds to relatively weak decay;
232
a 10-fold increase in y only reduces D by 50%.
3.5. Alongshore Dye Transport
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Dye is advected by the northward alongshore current V (Figures 2b and 3) from the
release location (x, y) = (−10, 0) m past the cross-shore array at yf = 248 m. The
alongshore dye transport T y,A/B from region A to region B through yf (Figure 3, diamonds)
is estimated for the surfzone,
y,A/B
TSZ (t)
=
Z
0
d(x, t)V (x, t)D(x, t) dx,
(2)
d(x, t)V (x, t)D(x, t) dx,
(3)
xb
and the inner-shelf,
y,A/B
TIS
(t)
=
Z
xb
xf6
233
using in situ, 30 s-averaged total water depth d = h + η, alongshore current V , and D at
234
f1-f6, assuming vertically uniform V (x, t) and D(x, t). For the surfzone, this assumption
235
is a good approximation. The vertical structure of surfzone dye concentration is measured
236
at the seaward surfzone boundary xb in mean water depth hb = 2.1 m, where three ETs on
237
f4 are located 0.2, 0.7, and 1.3 m above the bed (Figure 2c). As for a 29 Sept dye release
238
[Hally-Rosendahl et al., 2014], surfzone D is vertically uniform on 13 Oct (Figure 8).
239
However, inner-shelf D is not necessarily vertically uniform, as thermal stratification can
240
inhibit inner-shelf vertical dye mixing, even immediately offshore of the vertically mixed
241
surfzone (not shown here, similar to 29 Sept dye release in Hally-Rosendahl et al. [2014],
242
Figure 15). Inner-shelf V may also be vertically sheared, likely larger in the upper water
243
column, driven by southerly wind. Lastly, dye at yf sometimes extends offshore of f6
244
(e.g., Figure 3), but V measurements, and therefore the extent of cross-shore integration
245
for (3), are limited to xf6 = −135 m. For these reasons, TIS
246
247
y,A/B
is biased low.
y,A/B
and TIS
The surfzone and inner-shelf alongshore dye transports TSZ
y,A/B
generally vary
between approximately 0-1000 ppb m3 s−1 and 0-400 ppb m3 s−1 , respectively (Figures 9a
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y,A/B
248
and 9b). Averaged during the release period, T SZ
y,A/B
= 320 ppb m3 s−1 and T IS
=
249
76 ppb m3 s−1 , about 62% and 15% of the dye release rate Q, respectively. Therefore, 77%
250
of the released dye is alongshore-transported past yf = 248 m between the shoreline and
251
xf6 = −135 m (recall TIS
y,A/B
252
is biased low).
The cumulative (time-integrated) alongshore dye transports at yf for the surfzone and
Rt
y,A/B
T
t0 SZ
(τ ) dτ and
Rt
y,A/B
TIS
(τ ) dτ , respectively, where t0 = 10:39 h is the
253
inner-shelf are
254
dye release start time and TSZ
255
surfzone alongshore transport
256
ure 9c) with small steps corresponding to pulses of TSZ
257
after the last of the dye is advected past the cross-shore array, approximately four times
258
as much dye has been alongshore-transported through yf between the shoreline and f4 as
259
between f4 and f6 (Figure 9c).
y,A/B
Rt
t0
t0
y,A/B
and TIS
y,A/B
TSZ
are defined in (2) and (3). The cumulative
dτ is roughly linear during the dye release (Figy,A/B
(Figure 9a). At t ≈ 17:30 h,
4. Dye Mass Balances and Cross-shore Exchange
260
In sections 4.1-4.3, dye mass balances are shown to close in total, for near-field region
261
A and far-field region B, and for the surfzone and inner-shelf. These results are used in
262
section 4.4 to infer the surfzone to inner-shelf dye tracer exchange.
4.1. Mass Balance: Total and Regions
Surfzone and inner-shelf dye masses integrated over the entire alongshore domain (regions A (0 < y = 0 ≥ 248 m) and B (y > 248 m) combined) are
A+B
MSZ
(t)
=
Z
yp (t)
0m
D R A F T
Z
0m
xb
Z
0m
D(x, y, z, t) dz dx dy,
(4)
−h
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X - 15
and
A+B
MIS
(t)
=
Z
∞
−∞
Z
xb
−∞
Z
0m
D(x, y, z, t) dz dx dy,
(5)
−h
where yp (t) is the alongshore location of the leading edge of the northward-advecting dye
plume (green triangles in Figure 3). Estimation methods for (4) and (5) are described in
Appendix A. The total dye mass balance for the surfzone and inner-shelf is
A+B
MSZ
(t)
+
A+B
MIS
(t)
=
Z
t
Q dτ ,
(6)
t0
263
where Q is the steady dye release rate, t0 = 10:39 h is the dye release start time, and
264
A+B
A+B
MSZ
(t0 ) ≡ MIS
(t0 ) ≡ 0 ppb m3 .
265
The total dye mass balance (6) closes well over the 11:21-15:32 h time span of aerial
266
images (Figure 10a, compare red asterisks with line). On average, 88% of the released dye
267
tracer is accounted for using aerial and (relatively sparse) in situ measurements. Note,
268
A+B
B
MSZ
(and therefore MSZ
) may be biased low because yp (t) may be underestimated
269
(Appendix A1). Analyses are broken into time periods I, II, and III based on temporal
270
gaps in aerial data (e.g., Figure 10a) and dye plume evolution (recall section 3.2 and
271
A+B
A+B
Figure 3). Early in the release during period I, MSZ
≈ MIS
(Figure 10a). Starting in
272
A+B
period II, as more dye spreads from the surfzone to the inner-shelf, MIS
becomes larger
273
A+B
A+B
A+B
. In period III, MIS
≈ 2MSZ
(Figure 10a), indicating significant transport
than MSZ
274
of the surfzone-released dye to the inner-shelf.
275
The surfzone and inner-shelf are also decomposed into near-field region A and far-
276
A
B
field region B (Figure 3a). For the surfzone, MSZ
≈ MSZ
during period I (Figure 10b),
277
when dye has not advected very far downstream (e.g., Figure 3a). As the dye plume
278
B
A
advects farther alongshore during period II, MSZ
becomes larger than MSZ
. Though
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279
dye concentrations are highest near the release (region A) and decrease downstream, the
280
power-law decay is weak (equation (1) and Figure 7), and the larger alongshore extent of
281
B
A
the dye plume in region B than region A results in period II MSZ
≈ 2MSZ
(Figure 10b).
282
B
During period III, dye has advected far downstream (e.g., Figures 3e and 3f), and MSZ
≈
283
A
4MSZ
(Figure 10b). Similar trends are observed for the inner-shelf. During period I,
284
A
B
B
A
MIS
≈ MIS
(Figure 10c). In period II, MIS
begins to dominate MIS
, and during period
285
B
A
III, MIS
MIS
(Figure 10c).
4.2. Mass Balance: Near-field Region A
In near-field region A, the total released dye mass (
Rt
t0
Q dτ ) must balance the surfzone
A
A
and inner-shelf accumulated dye mass (MSZ
+ MIS
) and the time-integrated alongshore
R t y,A/B
y,A/B
transport from A to B ( t0 (TSZ
+ TIS
) dτ ), i.e.,
A
MSZ
(t)
+
A
MIS
(t)
+
Z t
t0
y,A/B
TSZ
+
y,A/B
TIS
dτ =
Z
t
Q dτ .
(7)
t0
286
On average, the sum of the observed mass and cumulative transports account for 76% of
287
the released dye (Figure 11, compare red asterisks with red line). The largest terms of (7)
288
are the cumulative surfzone and inner-shelf alongshore transports, with
289
0.5
290
A
A
The region A dye masses MSZ
and MIS
are relatively small, especially during period III
291
(Figure 11, triangles). The 24% of dye unaccounted for in region A is consistent with the
292
low-bias of TIS
Rt
t0
Q dτ and
Rt
t0
y,A/B
y,A/B
TIS
dτ ≈ 0.2
Rt
t0
Rt
t0
y,A/B
TSZ
dτ ≈
Q dτ (Figure 11, gray and blue curves, respectively).
(described in section 3.5).
4.3. Mass Balance: Far-field Region B
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Similar to the near-field region A, the far-field region B accumulated dye mass must
balance the A to B alongshore transport, i.e.,
B
MSZ
(t)
+
B
MIS
(t)
=
Z t
t0
y,A/B
y,A/B
TSZ
+ TIS
dτ.
(8)
293
The observed mass and transport estimates agree well with 18% relative rms error (Fig-
294
ure 12, compare red circles with squares), confirming the consistency among aerial and
295
in situ data and the validation of mass and transport estimation methods. On average,
296
the region B accumulated dye mass (red squares, Figure 12) is slightly larger than the
297
time-integrated alongshore transport (red circles, Figure 12), again consistent with the
298
low bias of TIS
y,A/B
.
4.4. Exchange Between the Surfzone and Inner-shelf
299
Because the section 4.1-4.3 dye mass balances close, cross-shore surfzone to inner-shelf
300
x,A
x,B
transport estimates for regions A and B (TSZ/IS
and TSZ/IS
, respectively) can be inferred
301
from observations. The region A inner-shelf dye mass must balance cross-shore transport
302
input from the region A surfzone and alongshore transport loss to the region B inner-shelf
303
(Figure 13). The region B inner-shelf dye mass must balance cross-shore transport input
304
from the region B surfzone and alongshore transport input from the region A inner-shelf
305
(Figure 13). The corresponding equations are
Z
t
Zt0t
t0
D R A F T
x,A
TSZ/IS
dτ =
A
MIS
(t)
+
x,B
B
(t) −
TSZ/IS
dτ = MIS
Z
t
Zt0t
t0
y,A/B
dτ ,
(9a)
y,A/B
dτ .
(9b)
TIS
TIS
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Adding (9a) and (9b), one recovers the expected inner-shelf balance for regions A and B
combined:
Z t
t0
x,A
x,B
TSZ/IS
+ TSZ/IS
A
B
dτ = MIS
(t) + MIS
(t) ,
(10)
306
stating that the total (A+B) inner-shelf-accumulated dye mass is the time integral of
307
the total cross-shore transport of surfzone-released dye. The time-integrated cross-shore
308
transports (9a), (9b), and (10) are inferred from the observed inner-shelf dye mass and
309
alongshore transport.
310
The inferred time-integrated cross-shore transports
Rt
t0
x
dτ are approximately linTSZ/IS
311
x
are estimated
ear in each time period, and the associated cross-shore transports TSZ/IS
312
x,A
from the slope of each best fit line (Figure 14). The region A cross-shore transport TSZ/IS
313
is similar for periods I and II (137 and 115 ppb m3 s−1 , respectively), consistent with the
314
fixed 248 m alongshore extent of region A, independent of the dye plume advecting far-
315
x,B
ther northward with time. In contrast, the region B cross-shore transport TSZ/IS
increases
316
significantly among periods I, II, and III (25, 263, and 495 ppb m3 s−1 , respectively) as
317
yp (t) moves northward (e.g., Figures 3 and 4). As a result, the region A and B combined
318
x,A+B
cross-shore transport TSZ/IS
also increases with time. During period I, when dye has not
319
x,A+B
advected far downstream (e.g., Figure 3a), TSZ/IS
= 162 ppb m3 s−1 = 0.32Q. During
320
x,A+B
period II, when dye has advected farther downstream (e.g., Figures 3b-3d), TSZ/IS
=
321
378 ppb m3 s−1 = 0.74Q. During period III, when dye has advected approximately 3 km
322
x,A+B
downstream (e.g., Figures 3e and 3f), TSZ/IS
= 498 ppb m3 s−1 = 0.97Q, and most of
323
the cross-shore transport occurs in region B (Figure 14, compare red with green best fit
324
slopes). Over the approximate 5 h of aerial observations, roughly 1/2 of the shoreline-
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325
released dye is cross-shore transported to the inner-shelf, i.e.,
326
(Figure 14, compare blue symbols with thin red line).
Rt
t0
x,A+B
TSZ/IS
dτ ≈
1
2
Rt
t0
Q dτ
5. Discussion
5.1. Parameterizing Cross-Shore Transport Estimates
The box-model-based cross-shore tracer flux parameterization used for temperature by
Hally-Rosendahl et al. [2014] is tested here for dye with the inferred estimates of surfzone
x
to inner-shelf cross-shore dye transport (section 4.4). The cross-shore dye flux FˆSZ/IS
(units ppb m2 s−1 ) at the surfzone/inner-shelf boundary xb is parameterized as
x
FˆSZ/IS
= hb u∗ ∆D,
(11)
where hb is the water depth at xb , u∗ is a bulk cross-shore exchange velocity, and
∆D = DSZ − DIS is the surfzone to inner-shelf mean dye concentration difference. Here
∆D is computed separately for region A and region B and for each time period using
A
A
B
B
period-averaged dye mass estimates M SZ , M IS , M SZ , and M IS (section 4.1) and apA
A
A
). The surfzone volumes are
proximate volumes V of each region (e.g., DSZ = M SZ /VSZ
A,B
defined by the integration regions for M SZ (see Appendix A1). The inner-shelf volumes
are estimated using hdye (Appendix A2), cross-shore width | − 250 m − xb | = 169 m
(e.g., Figure 3), and alongshore extents yf = 248 m and y p − yf for regions A and B, respectively, where y p is the mean of yp for each time period . The parameterized surfzone
x
to inner-shelf cross-shore dye transports TˆSZ/IS
(units ppb m3 s−1 ) are then
x,A
TˆSZ/IS
x,B
TˆSZ/IS
=
D R A F T
Z
yp
yf
=
Z
yf
0
x,A
FˆSZ/IS
x,B
FˆSZ/IS
dy =
dy =
Z
Z
yf
A
A
hb u∗ ∆D dy = hb u∗ ∆D yf ,
(12a)
0
yp
B
hb u∗ ∆D dy = hb u∗ ∆D
yf
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B
y p − yf
(12b)
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for region A and region B, respectively, and
x,A+B
x,A
x,B
TˆSZ/IS
= TˆSZ/IS
+ TˆSZ/IS
(13)
327
x,A
x,B
x,A+B
for regions A and B combined. The parameterized TˆSZ/IS
, TˆSZ/IS
, and TˆSZ/IS
are each
328
computed for periods I, II, and III.
329
330
x
x
Parameterized TˆSZ/IS
and inferred TSZ/IS
are generally similar (Figure 15) with squared
correlation r2 = 0.85 and best fit slope 0.7. Minimizing the rms error among parameterized
331
x
x
TˆSZ/IS
and inferred TSZ/IS
transports yields the best fit cross-shore exchange velocity
332
u∗ = 1.2 × 10−2 m s−1 . This is consistent with the u∗ = 0.9 × 10−2 m s−1 found using
333
temperature observations on another day with similar wave conditions [Hally-Rosendahl
334
et al., 2014].
335
x,A
The parameterized TˆSZ/IS
are comparable between periods I and II and are similar to
336
x,A
the inferred TSZ/IS
(Figure 15, green circle and square), while the period III parameterized
337
x,A
x,A
TˆSZ/IS
over-estimates the small inferred TSZ/IS
(Figure 15, green triangle). The param-
338
x,B
eterized TˆSZ/IS
increases among periods I, II, and III (Figure 15, vertical coordinates of
339
red symbols) as yp (t) moves farther northward (Figure 4), consistent with the increase
340
x,B
of inferred TSZ/IS
(Figure 15, horizontal coordinates of red symbols). Similarly, parame-
341
x,A+B
x,A+B
terized TˆSZ/IS
and inferred TSZ/IS
are comparable and both increase with time among
342
periods I, II, and III (Figure 15, blue symbols).
343
x,A
During period III, the inferred TSZ/IS
is small (horizontal coordinate of green triangle
344
in Figure 15), suggesting that most of the region A surfzone dye is transported alongshore
345
to the region B surfzone rather than offshore to the inner-shelf. Consistent with this
346
inference, the mean period III surfzone alongshore transport T SZ
y,A/B
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= 507 ppb m3 s−1 is
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347
X - 21
within 1% of the dye release rate Q = 512 ppb m3 s−1 . Combined with small period III
d
dt
A
MIS
(Figure 10c, triangles), this confirms that the period III cross-shore
348
A
MIS
and
349
transport within region A is small.
350
The inferred cross-shore exchange velocity u∗ = 1.2 × 10−2 m s−1 incorporates all po-
351
tential surfzone/inner-shelf exchange mechanisms, including both rip currents and Stokes
352
drift driven exchange (onshore near-surface mass flux balanced by undertow at depth).
353
Clark et al. [2010] found the Stokes drift driven exchange mechanism to be negligible for
354
cross-shore dye dispersion within the surfzone due to the surfzone being vertically well-
355
mixed (assumed in Clark et al. [2010], demonstrated here in Figure 8). However, seaward
356
of the surfzone, where dye concentration is not necessarily vertically uniform, the Stokes
357
drift driven exchange mechanism could potentially be important. Here, the u∗ magnitude
358
is compared to the estimated Stokes drift driven exchange velocity offshore of the surfzone.
359
Assuming normally incident, narrow-banded waves with the observed Hs (Figure 2a) and
360
peak period Tp = 13 s, the Lagrangian (Eulerian plus Stokes) velocity profile [e.g., Hally-
361
Rosendahl et al., 2014] at f6 is shoreward in the upper water column and seaward only
362
below z = −1.8 m. Averaging vertically over the seaward portion of the velocity profile
363
yields uLsea = 5.9 × 10−4 m s−1 , which is 20× smaller than the inferred exchange velocity
364
u∗ = 1.2 × 10−2 m s−1 . Observations at x ≈ 2xb (roughly 25 m offshore of f6) show that
365
inner-shelf dye (which has a vertically-mixed surfzone source) is surface-intensified and
366
decreases with depth (Figure 16). This requires seaward dye advection in the upper water
367
column, inconsistent with the Stokes drift driven exchange mechanism (shoreward in the
368
upper water column and seaward at depth). Furthermore, inner-shelf D is distinctively
369
alongshore-patchy (Figures 3 and 13), but Stokes drift driven exchange is expected to be
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370
approximately alongshore-uniform. These discrepancies in magnitude, vertical structure,
371
and alongshore-patchiness suggest that the observed surfzone to inner-shelf cross-shore
372
dye transport is dominated by transient rip ejections on this day with moderate waves, as
373
also concluded using temperature observations from another day with similar wave condi-
374
tions [Hally-Rosendahl et al., 2014]. Farther seaward of 2xb , Stokes drift driven exchange
375
may be important [e.g., Lentz et al., 2008; Suanda and Feddersen, 2015].
376
x,A
The small TSZ/IS
inferred for region A during period III and the associated lack of
377
significant transient rip ejections is not surprising, as transient rip currents are sporadic
378
in space and time, and region A is small (< 250 m alongshore) and period III short
379
(< 40 min). The cross-shore transport parameterization (12a) is a bulk quantity that
380
does not resolve the spatial or temporal variability of individual transient rip ejections.
381
Therefore, as ∆DA is comparable among periods I, II, and III, the parameterization (12a)
382
x,A
is unable to reproduce the smaller period III inferred TSZ/IS
. However, the period III
383
x,A+B
parameterized TˆSZ/IS
for the entire alongshore domain (regions A and B combined) still
384
x,A+B
(Figure 15, blue triangle).
agrees well with the inferred TSZ/IS
6. Summary
385
A continuous 6.5 h, near-shoreline release of fluorescent Rhodamine WT dye tracer
386
was observed on 13 October, 2009, at the alongshore-uniform Imperial Beach, California
387
(IB09 experiment). Surfzone and inner-shelf dye concentration was measured in situ with
388
fixed and mobile (jetski- and boat-mounted) fluorometers, and remotely with an aerial
389
multispectral camera system. Waves and currents were measured between the shoreline
390
and roughly 4 m water depth. Dye was advected alongshore by breaking wave- and wind-
391
driven currents, forming a several km-long plume.
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392
Aerial images showed the plume advecting alongshore while ≈ 50 m-wide dye bands
393
were ejected from the surfzone to the inner-shelf by transient rip currents. Near the
394
shoreline, alongshore dye dilution over ≈ 2 km was relatively weak and followed a power-
395
law with exponent −0.3, similar to 29 September dilution over 700 m with exponent −0.2
396
[Hally-Rosendahl et al., 2014]. Both of these dilution rates were smaller than previously
397
observed and modeled for Fickian dilution of surfzone-contained dye plumes (exponent
398
≈ −0.5, Clark et al. [2010, 2011]). At a cross-shore instrument array 248 m downstream
399
of the release location, ≈ 75% of the released dye was alongshore-advected through the
400
array within 135 m of the shoreline.
401
Using aerial and in situ measurements, 88% of the released dye mass was accounted for
402
over the entire domain (≈ 400 m cross-shore and 3 km alongshore) during a 5 h period after
403
the start of the release. Mass and alongshore transport observations for separate near-
404
and far-field regions were also in agreement (and the small discrepancies were consistent
405
with low-biased inner-shelf alongshore dye transport estimates). The closure of these dye
406
mass balances allowed for inferred estimates of surfzone to inner-shelf cross-shore dye
407
transport, which amounted to roughly 1/2 of the shoreline-released dye during this 5 h
408
period.
409
The cross-shore dye transport was parameterized well using a bulk exchange velocity and
410
surfzone/inner-shelf mean dye concentration difference (inferred and parameterized cross-
411
shore dye transports were correlated (r2 = 0.85) with best fit slope 0.7). The resulting
412
best fit bulk exchange velocity was u∗ = 1.2 × 10−2 m s−1 , consistent with a temperature-
413
derived u∗ = 0.9×10−2 m s−1 from another day of IB09 with similar waves. Together with
414
the observed inner-shelf dye alongshore-patchiness and surface intensification, a 20-fold
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415
difference between u∗ and Stokes drift driven velocity magnitudes indicates that transient
416
rip currents were the dominant surfzone to inner-shelf tracer exchange mechanism at this
417
alongshore-uniform beach during moderate wave conditions.
Appendix A: Dye Mass Estimates
A1. Surfzone
The total surfzone dye mass in regions A and B is defined as
A+B
MSZ
(t)
Z
=
yp (t)
0m
Z
0m
xb
Z
0m
D(x, y, z, t) dz dx dy,
(A1)
−h
where h is the water depth, xb is the seaward surfzone boundary, and yp (t) is the leading
alongshore edge of the northward-advecting dye plume (Figures 3 and 4). The surfzone dye
mass is estimated at times corresponding to the aerial images. The three MSZ integrals
(dz, dx, dy) are estimated as follows. Dye is vertically well mixed in the surfzone (Figure 8
and Hally-Rosendahl et al. [2014]), and therefore the vertical integral becomes
Z
0m
D(x, y, z, t) dz = hD(x, y, t).
(A2)
−h
Cross-shore D profiles do not exist for all y and t. However, cross-shore jetski transects
were repeated at various y and are used to compute time-averaged cross-shore dye profiles
(see section 3.3 and Figure 5) at alongshore locations near f2 and SA1-SA4, where nearshoreline dye was measured continuously. These mean profiles are used to compute a
surfzone dye cross-shore uniformity parameter ξ(y) defined as
ξ(y) =
R0m
xb
h(x)D(x, y)dx
hSZ LSZ D(x ≈ −10 m, y)
,
(A3)
where hSZ is the mean surfzone water depth, LSZ = |xb | is the surfzone width, and
x ≈ −10 m is the location of the shoreward-most observations. By definition 0 < ξ(y) ≤ 1,
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with ξ ≈ 0 corresponding to shoreline-released dye being highly shoreline-concentrated
and ξ = 1 corresponding to dye being perfectly cross-shore uniform. Close to the dye
release (y = 14 m), the cross-shore uniformity parameter ξ ≈ 0.5. Downstream, as dye
mixes across the surfzone, ξ increases to > 0.9 by y = 810 m (Figure 17). The cross-shore
integral is estimated as
Z
0m
h(x) D(x, y, t) dx = hSZ LSZ ξ(y)Dsl (y, t),
(A4)
xb
where Dsl (y, t) is dye measured near the shoreline. Combining (A1)-(A4) yields
A+B
MSZ
(t)
=
Z
yp (t)
hSZ LSZ ξ(y)Dsl (y, t) dy.
(A5)
0
418
The integral (A5) is alongshore-integrated numerically using the trapezoid rule with
419
Dsl (y, t) at y = 1 m, ySA1 , yf , ySA2 , ySA3 , ySA4 , and yp (t) (Figure 3, yellow symbols
420
and green triangle). Nearest the release, Dsl (y = 1 m, t) = 98 ppb is estimated via the
421
best-fit (1) (see Figure 7) during times that dye is being released (note that all M esti-
422
mates are during this time period). Downstream, Dsl (yp (t), t) is also estimated using the
423
best-fit (1) (see Figures 4 and 7).
424
A+B
A
B
MSZ
(t) are decomposed into MSZ
(t) (0 < y ≤ yf ) and MSZ
(t) (yf < y ≤ yp (t))
425
using the alongshore boundary yf = 248 m. MSZ estimates are computed for all aerial
426
B
image times when in situ near-shoreline dye data are available. Note that MSZ
(t) and
427
A+B
(t) may be biased low because the inner-shelf yp (t) may be smaller than the actual
MSZ
428
alongshore extent of the dye plume within the surfzone (where the alongshore current is
429
fastest (Figure 2b)) and the transient rip ejections from the surfzone to the inner-shelf
430
are episodic in space and time.
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A2. Inner-Shelf
The total inner-shelf dye mass in regions A and B is defined as
A+B
MIS
(t)
=
Z
Z
∞
Z
xb
−∞
−∞
0m
D(x, y, z, t) dz dx dy,
(A6)
−h
where h is the water depth, and xb is the surfzone/inner-shelf boundary. Inner-shelf dye
mass estimates are calculated using surface dye concentration maps Ds (x, y, t) from the
aerial images (e.g., Figure 3) and in situ observations of inner-shelf vertical dye structure
(Figure 16) from boat-towed vertical array data (section 2.3.4). These vertical array data
resolve inner-shelf D for z = −1 to −3 m (section 2.3.4), thus requiring assumptions for
the vertical structure outside this range. As inner-shelf dye comes from the vertically
mixed surfzone (Figure 8 and Hally-Rosendahl et al. [2014]), inner-shelf D(x, y, z, t) is
assumed vertically uniform in the upper 1 m. For z < −3 m, the best fit of mean D(z)
(Figure 16) is extrapolated to the depth where it would vanish. This structure is then
vertically integrated, and hdye is computed as the depth that yields an equivalent vertical
integral hdye Ds (x, y, t). The inner-shelf vertical dye integral is thus estimated as
Z
0
D(x, y, z, t) dz = hdye Ds (x, y, t),
(A7)
−h
where h is the water depth, Ds (x, y, t) is the aerial-measured surface dye concentration,
and hdye = min (2.67 m, h). The inner-shelf dye mass estimates are then
A+B
MIS
(t)
=
Z
∞
−∞
Z
xb
hdye Ds (x, y, t) dx dy,
(A8)
−∞
431
A+B
integrated using the trapezoid rule in each lateral direction. MIS
(t) are decomposed
432
A
B
into MIS
(t) (−∞ < y ≤ yf ) and MIS
(t) (yf < y < ∞) using the alongshore boundary
433
yf = 248 m.
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434
Acknowledgments.
435
IB09 field work and analysis was funded by ONR, NSF, and California Sea Grant.
436
K. Hally-Rosendahl was supported by the National Science Foundation Graduate Re-
437
search Fellowship under Grant No. DGE1144086 and California Sea Grant under Project
438
No. R/CONT-207TR. Staff and students from the Integrative Oceanography Division
439
(B. Woodward, B. Boyd, K. Smith, D. Darnell, R. Grenzeback, A. Gale, M. Spydell,
440
M. Omand, M. Yates, M. Rippy, A. Doria) were instrumental in acquiring the field obser-
441
vations. K. Millikan, D. Ortiz-Suslow, M. Fehlberg, and E. Drury provided field assistance.
442
Imperial Beach lifeguards, supervised by Captain R. Stabenow, helped maintain public
443
safety. The YMCA Surf Camp management generously allowed extensive use of their
444
facility for staging and recuperation. The U. S. Navy provided access to Naval property
445
for data collection. Dr. M. Okihiro coordinated permits and logistics. We thank these
446
people and organizations. This work comprises a portion of K. Hally-Rosendahl’s PhD
447
thesis. As subsequent thesis chapters will use these same IB09 data, they are not shared
448
with this work. However, upon completion of K. Hally-Rosendahl’s thesis, IB09 data will
449
be made available at http://falk.ucsd.edu in accordance with the AGU data policy.
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
S A4
boat
1600
1400
1200
S A3
800
b e ac h
y ( m)
1000
je t s ki
600
S A2
400
f6
f4
f2
200
S A1
0
−250
Figure 1.
−200
−150
−100
x ( m)
−50
0
Planview of IB09 bathymetry contours versus cross-shore coordinate x and
alongshore coordinate y. Star indicates dye release location. Diamonds denote the crossshore array of bottom-mounted instrument frames f1-f6 (onshore to offshore). Circles
indicate SA1-SA4 fluorometer locations. Vertical dashed line represents an idealized boat
alongshore transect driven repeatedly near this cross-shore location. Horizontal dashed
line represents an idealized jetski cross-shore surface transect driven repeatedly at various
alongshore locations.
D R A F T
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
H s ( m)
1
( a)
0.5
V ( m s − 1)
0
0.4
xb
(b)
0.2
0
z ( m)
0
xb
(c)
f3
−2
−4
−150
f2
f1
f4
f5
f6
−100
xb
−50
x ( m)
0
Figure 2. Time-averaged (11:00-16:00 h) (a) significant wave height Hs , (b) alongshore
current V , and (c) vertical locations of f1-f6 versus cross-shore coordinate x. In (c), the
black curve gives the bathymetry h(x). The mean seaward surfzone boundary xb = −81 m
is defined as the location of maximum Hs .
D R A F T
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
Figure 3.
Aerial multispectral images of surface dye concentration (see colorbar)
versus cross-shore coordinate x and alongshore coordinate y for six times (indicated in
each panel). The mean shoreline is at x = 0 m. Green star indicates location of continuous
dye release (starting at t0 = 10:39 h). Yellow diamonds indicate cross-shore array f1-f6
locations, and yellow circles indicate SA1-SA4 locations. Light gray indicates regions
outside the imaged area, and black indicates unresolved regions due to foam from wave
breaking. Vertical dashed cyan line at xb divides the surfzone (SZ) and inner-shelf (IS),
and horizontal cyan line divides the near- and far-field regions A and B (see panel (a)).
Plume leading edge yp (t) is shown with green triangles at x ≈ −100 m (for panel (f),
yp ≈ 3250 m). Panels (a), (b)-(d), and (e)-(f) are in time periods I, II, and III, respectively.
D R A F T
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
3500
3000
y p ( m)
2500
2000
1500
1000
500
I
0
11
Figure 4.
II
12
III
13
14
t (h)
15
16
17
Alongshore coordinate of dye plume leading edge yp versus time. The
determination of yp (t) is described in section 3.2. Magenta bar indicates duration of nearshoreline, continuous dye release at y = 0 m. Blacks bars denote time periods I, II, and
III.
100
D (ppb)
80
y
y
y
y
y
=
=
=
=
=
14 m
87 m
207 m
619 m
810 m
60
40
20
0
−80
Figure 5.
−60
−40
x ( m)
−20
0
Time-averaged, cross- and alongshore-binned surfzone D from jetski surface
transects versus cross-shore coordinate x (see legend for alongshore locations y).
D R A F T
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
200
( a) S A1, y = 82 m
D (ppb)
150
100
50
0
80
( b ) f 2, y = 248 m
D (ppb)
60
40
20
0
50
( c ) S A2, y = 546 m
D (ppb)
40
30
20
10
0
20
( d ) S A3, y = 1069 m
D (ppb)
15
10
5
0
20
( e ) S A4, y = 1662 m
D (ppb)
15
10
5
0
10
Figure 6.
11
12
13
14
15
t (h)
16
17
18
19
20
Dye concentration D versus time at the near-shoreline f2 and SA1-SA4
(diamond and circles, respectively, in Figures 1 and 3). Alongshore location is indicated
in each panel. Magenta bars indicate duration of near-shoreline, continuous dye release at
y = 0 m. Vertical axes differ, and each time series begins at the instrument deployment
time (not dye arrival time).
D R A F T
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
2
D (ppb)
10
1
10
0
10 0
10
Figure 7.
2
10
y ( m)
4
10
Mean (time-averaged) dye concentration versus alongshore coordinate y at
the near-shoreline f2 and SA1-SA4 (yellow diamond and circles in Figure 3, respectively).
Vertical bars are standard deviations about the means. Best fit line (dashed) is D =
D0 (y/y0 )α , where y0 = 1 m is chosen for simplicity, and best fit constants are D0 = 98 ppb
and α = −0.3.
D R A F T
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
50
D (ppb)
f 4, 1. 3 mab
40
f 4, 0. 7 mab
30
f 4, 0. 2 mab
20
10
0
10
Figure 8.
11
12
13
14
t (h)
15
16
17
18
Dye concentration D versus time at three ETs with different vertical ele-
vations (mab is meters above bottom) on f4 at the seaward surfzone boundary xb (see
legend and Figure 2c). Magenta bar indicates duration of near-shoreline, continuous dye
release (Figure 1, star) 248 m south of the cross-shore array (Figure 1, diamonds). Gaps
in the time series result from sampling for 51 minutes of each hour.
D R A F T
March 17, 2015, 12:07pm
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
TSZ
A/B , y
( p p b m 3 s − 1)
4000
SZ
( a)
IS
(b)
SZ
IS
(c)
3000
2000
1000
( p p b m 3 s − 1)
0
800
600
TI S
A/B , y
400
200
6
4
2
Rt
t0 T
A/B , y
3
d τ (ppb m )
6
0
x 10
8
0
10
Figure 9.
11
12
13
14
t (h)
15
16
17
18
Time series of alongshore dye transport from region A to B in the (a)
y,A/B
surfzone (TSZ
y,A/B
defined in (2)) and (b) inner-shelf (TIS
defined in (3)). Vertical scales
differ. (c) Time series of cumulative (time-integrated) surfzone and inner-shelf alongshore
dye transports (see legend). Magenta bars indicate duration of near-shoreline, continuous
dye release (Figure 3a, star) 248 m south of the cross-shore array (Figure 3a, diamonds)
dividing regions A and B . The dye release rate Q = 512 ppb m3 s−1 , and the total dye
released is 1.19 × 107 ppb m3 .
D R A F T
March 17, 2015, 12:07pm
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
6
x 10
10
Rt
II
III
t0 Q d τ
MSZ + MI S
MSZ
MI S
8
M ( p p b m 3)
I
6
( a)
4
2
6
0
x 10
10
(b)
Rt
(c)
t0 Q d τ
M SAZ + M SBZ
M SAZ
M SBZ
8
M ( p p b m 3)
Rt
6
4
2
6
0
x 10
10
M ( p p b m 3)
8
6
t0 Q d τ
M IAS + M IBS
M IAS
M IBS
4
2
0
11
Figure 10.
11.5
12
12.5
13
13.5
t (h)
14
14.5
15
15.5
16
Dye mass M versus time. Black bars denote time periods I, II, and III.
(a) Surfzone estimates MSZ (gray) are from in situ observations, and inner-shelf estimates
MIS (blue) are from aerial images. Red asterisks are MSZ +MIS . Red line shows the timeRt
integrated dye mass released since t0 =10:39 h ( t0 Q dτ , where Q is the dye release rate).
Estimation methods for MSZ and MIS are described in Appendix A1 and A2, respectively.
(b) Surfzone dye mass MSZ versus time for the near-field region A (y ≤ 248 m, triangles)
A+B
and the far-field region B (y > 248 m, squares). Solid gray diamonds are MSZ
. (c)
Inner-shelf dye mass MIS versus time for region A (triangles) and region B (squares).
A+B
Solid blue circles are MIS
.
D R A F T
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
6
x 10
10
Rt
II
III
t0 Q d τ
2
R t 1 y , A/B
y , A/B
d τ + M SAZ ( t ) + M IAS ( t )
TSZ
+ TI S
t0
Rt
y , A/B
TSZ
dτ
Rtt0 y , A/B
dτ
t0 T I S
A
MSZ (t)
M IAS ( t )
8
M ( p p b m 3)
I
6
4
2
0
11
11.5
12
12.5
13
13.5
t (h)
14
14.5
15
15.5
16
Dye mass balance terms versus time for near-field region A (0 < y ≤
Figure 11.
yf = 248 m). See legend and equation (7). Red line shows the time-integrated dye mass
released since t0 =10:39 h.
6
x 10
10
I
Rt 1
t0
M ( p p b m 3)
8
6
4
II
y , A/B
y , A/B
+ TI S
TSZ
M SBZ ( t ) + M IBS
Rt
y , A/B
dτ
t0 T S Z
Rt
y , A/B
dτ
t0 T I S
B
MSZ (t)
M IBS ( t )
III
2
dτ
(t)
2
0
11
11.5
12
12.5
13
13.5
t (h)
14
14.5
15
15.5
16
Figure 12. Dye mass balance terms versus time for far-field region B (y > yf = 248 m).
See legend and equation (8).
D R A F T
March 17, 2015, 12:07pm
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
SZ
IS
B
MIS
R
B
R
A
x,B
TSZ/IS
d⌧
y,A/B
TIS
R
A
MIS
d⌧
x,A
TSZ/IS
d⌧
R
Q d⌧
Figure 13. Planview photograph and superposed schematic of dye mass balances (9a),
(9b), and (10). Star denotes location of dye released at steady rate Q.
D R A F T
March 17, 2015, 12:07pm
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
6
x 10
Rt
x
t 0 T S Z /I S
d τ ( p p b m3 )
10
8
6
Rt
I
II
III
Q dτ
2
Rtt0 1 x , A
x, B
T
+
T
S Z /I S
S Z /I S d τ
t
R t0 x , A
T
dτ
Rtt0 SxZ, B/I S
t 0 T S Z /I S d τ
4
2
0
11
11.5
12
12.5
13
13.5
t (h)
14
14.5
15
15.5
16
Figure 14. Time series of cumulative (time-integrated) cross-shore dye transports from
the surfzone to inner-shelf (circles) inferred from inner-shelf dye mass observations MIS
y,A/B
and alongshore transport measurements TIS
. See (9a), (9b), and (10). Line segments
are least squares fits for each time period, and line segment slopes yield inferred crossx
. Thin red line shows the time-integrated dye mass released
shore dye transports TSZ/IS
since t0 = 10:39 h.
D R A F T
March 17, 2015, 12:07pm
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
500
A+B
A
B
TˆSxZ /I S ( p p b m 3 s − 1 )
400
I
II
III
300
200
100
0
0
Figure 15.
100
200
300
400
T SxZ /I S ( p p b m 3 s − 1 )
500
x
versus inferred cross-shore
Parameterized cross-shore dye transport TˆSZ/IS
x
dye transport TSZ/IS
for regions A, B, and A+B during periods I, II, and III (see legend).
x
x
are inferred from
follow (12a), (12b), and (13). The TSZ/IS
The parameterized TˆSZ/IS
aerial and in situ observations (see (9a), (9b), and (10)).
D R A F T
March 17, 2015, 12:07pm
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HALLY-ROSENDAHL ET AL.: SURFZONE TO INNER-SHELF TRACER EXCHANGE
−1
z ( m)
−1.5
−2
−2.5
−3
0
5
D (ppb)
10
Figure 16. Mean inner-shelf dye concentration D versus vertical coordinate z from the
boat-towed vertical array for data within inner-shelf dye patches (D(x, y, z = −1 m, t) ≥
2 ppb). Dashed curves indicate standard deviations about the mean.
1
0.8
ξ ( )
0.6
0.4
0.2
0
0
Figure 17.
200
400
600
y (m )
800
1000
1200
Surfzone dye cross-shore uniformity parameter ξ versus alongshore coordi-
nate y (ξ is defined in (A3) and described in Appendix A1).
D R A F T
March 17, 2015, 12:07pm
D R A F T